Simplify and expand the following expression: $ \dfrac{-4}{3r - 10}+\dfrac{5r}{5r - 7} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(3r - 10)(5r - 7)$ Multiply the first term by $\dfrac{5r - 7}{5r - 7}$ $ \begin{align*} \dfrac{-4}{3r - 10} \times \dfrac{5r - 7}{5r - 7} & = \dfrac{(-4)(5r - 7)}{(3r - 10)(5r - 7)} \\ & = \dfrac{-20r + 28}{(3r - 10)(5r - 7)}\end{align*} $ Multiply the second term by $\dfrac{3r - 10}{3r - 10}$ $ \begin{align*} \dfrac{5r}{5r - 7} \times \dfrac{3r - 10}{3r - 10} & = \dfrac{(5r)(3r - 10)}{(5r - 7)(3r - 10)} \\ & = \dfrac{15r^2 - 50r}{(5r - 7)(3r - 10)}\end{align*} $ Now we have: $ = \dfrac{-20r + 28}{(3r - 10)(5r - 7)} + \dfrac{15r^2 - 50r}{(5r - 7)(3r - 10)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{-20r + 28 + 15r^2 - 50r}{(3r - 10)(5r - 7)} $ $ = \dfrac{-70r + 28 + 15r^2}{(3r - 10)(5r - 7)}$ Expand the denominator: $ = \dfrac{-70r + 28 + 15r^2}{15r^2 - 71r + 70}$